A079296 Primes ordered by decreasing value of the function p -> sqrt(q) - sqrt(p) where q is the next prime after p.

7, 113, 23, 13, 31, 3, 1327, 19, 47, 199, 139, 89, 5, 211, 293, 53, 523, 317, 61, 181, 73, 887, 1129, 83, 37, 241, 2, 43, 283, 1669, 11, 467, 1069, 337, 509, 2477, 131, 2179, 2971, 1259, 773, 1951, 1637, 409, 3271, 421, 151, 1381, 67, 839, 619, 863, 157, 17, 661, 3137

Offset: 1

Author

Thomas Nordhaus, Feb 09 2003

Keywords

• nonn
• nice

Comments

I computed a couple of thousand primes with EXCEL and ordered them accordingly. There is a very small chance that very large prime numbers will change the order of the given terms above.

This sequence only makes sense if the sequence n -> sqrt(p_(n+1)) - sqrt(p_n) is a zero-sequence which is a hard unsolved problem. See also Andrica's conjecture.

For each consecutive prime pair p < q, the number d = sqrt(q) - sqrt(p) is unique. Place d in order from greatest to least and specify p. See Table II in Wolf. A rearrangement of the primes. - Robert G. Wilson v, Oct 18 2012

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• C. K. Caldwell, Gaps between primes.
• Eric W. Weisstein, Andrica's Conjecture
• Wikipedia, Andrica's conjecture
• Marek Wolf, A note on the Andrica conjecture, arXiv:1010.3945 [math.NT], 2010.

Crossrefs

Cf. A078692, A002386, A084974 (records).

Programs

Mathematica

lim = 1/5; lst = {}; p = 2; q = 3; While[p < 50000, If[ Sqrt[q] - Sqrt[p] > lim, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; First@ Transpose@ Sort[lst, #1[[2]] > #2[[2]] &] (* Robert G. Wilson v, Oct 18 2012 *)

Extensions

More terms from Robert G. Wilson v, Oct 18 2012